Matrix of a linear transformation

Consider the following problem.

Let M denote the set of 2x2 real matrices. Let A be an element of M with trace 2 and determinant -3. Identifying M with R^{4}, consider the linear transformation T: M -> M defined by T(B) = AB. Then which of the following statements are true?

a) T is diagonalizable. b) 2 is an eigenvalue of T c) T is invertible d) T(B) = B for some non-zero matrix B in the set M.

Based on the opinion that the **matrix **of a linear transformation is the **matrix **which is multiplied by the **input matrix **of the domain, to get the **output matrix **belonging to the co-domain.

Here the definition of the transformation gives us the impression that it is obvious that the matrix A is the matrix of the linear transformation. So as per the given information since determinant of A is not zero it is invertible, which means it is also diagonalizable. Therefore options a and c are true. Since the trace is 2 and the determinant is -3, the two eigen values are -3 and +1 which shows that option b is not true. Similarly, since A cannot be the identity matrix, T(B) = AB can never be B itself. This means option d is also not true.

**My question is this.**

What does the phrase, **"Identifying M with R**^{4}" have to do with the problem?

Is there something overlooked by me due to my ignoring of the significance of this phrase?

Re: Matrix of a linear transformation

"Identifying M with $\displaystyle R^4$" means thinking of $\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix}$ as being the same as (a, b, c, d) with the "usual" addition and scalar multiplication. That effectively means $\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix}+ \begin{bmatrix}e & f \\ g & h\end{bmatrix}= \begin{bmatrix}a+ e & b+ f \\ c+ g & d+ h\end{bmatrix}$ and $\displaystyle \alpha\begin{bmatrix}a & b \\ c & d \end{bmatrix}= \begin{bmatrix}\alpha a & \alpha b \\ \alpha c & \alpha d \end{bmatrix}$.

Re: Matrix of a linear transformation

Thank you HallsofIvy... I sort of assumed it to be a default info... do you find any other mistake with the conclusions i've arrived at the options of the givenquestion?